Conformal transformation with multiple scalar fields and geometric property of field space with Einstein-like solutions

Abstract

Yong Tang and Yue-Liang Wu School of Astronomy and Space Sciences, University of Chinese Academy of Sciences (UCAS), Beijing, China School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China International Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China Abstract Multiple scalar fields appear in vast modern particle physics and gravity models. When they couple to gravity non-minimally, conformal transformation is utilized to bring the theory into Einstein frame. However, the kinetic terms of scalar fields are usually not canonical, which makes analytic treatment difficult. Here we investigate under what conditions the theories can be transformed to the quasi-canonical form, in which case the effective metric tensor in field space is conformally flat. We solve the relevant nonlinear partial differential equations for arbitrary number of scalar fields and present several solutions that may be useful for future phenomenological model building, including the σ-model with a particular non-minimal coupling. We also find conformal flatness can always be achieved in some modified gravity theories, for example, Starobinsky model.